Equations in Physics

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Equations in Physics

By ir. J.C.A. Wevers


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Contents

Contents I

Physical Constants 1

1 Mechanics 2

1.1 Point-kinetics in a fixed coordinate system . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.1.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.1.2 Polar coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2 Relative motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.3 Point-dynamics in a fixed coordinate system . . . . . . . . . . . . . . . . . . . . . . . . 2

1.3.1 Force, (angular)momentum and energy . . . . . . . . . . . . . . . . . . . . . . . 2

1.3.2 Conservative force fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.3.3 Gravitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.3.4 Orbital equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.3.5 The virial theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.4 Point dynamics in a moving coordinate system . . . . . . . . . . . . . . . . . . . . . . 4

1.4.1 Apparent forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.4.2 Tensor notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.5 Dynamics of masspoint collections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.5.1 The center of mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.5.2 Collisions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.6 Dynamics of rigid bodies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.6.1 Moment of Inertia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.6.2 Principal axes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.6.3 Time dependence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.7 Variational Calculus, Hamilton and Lagrange mechanics . . . . . . . . . . . . . . . . . 7

1.7.1 Variational Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.7.2 Hamilton mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.7.3 Motion around an equilibrium, linearization . . . . . . . . . . . . . . . . . . . . 7

1.7.4 Phase space, Liouville’s equation . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.7.5 Generating functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2 Electricity & Magnetism 9

2.1 The Maxwell equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.2 Force and potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.3 Gauge transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.4 Energy of the electromagnetic field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.5 Electromagnetic waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.5.1 Electromagnetic waves in vacuum . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.5.2 Electromagnetic waves in mater . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.6 Multipoles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.7 Electric currents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.8 Depolarizing field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.9 Mixtures of materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

I


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3 Relativity 133.1 Special relativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3.1.1 The Lorentz transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3.1.2 Red and blue shift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143.1.3 The stress-energy tensor and the field tensor . . . . . . . . . . . . . . . . . . . 14

3.2 General relativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143.2.1 Riemannian geometry, the Einstein tensor . . . . . . . . . . . . . . . . . . . . . 143.2.2 The line element . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.2.3 Planetary orbits and the perihelium shift . . . . . . . . . . . . . . . . . . . . . 163.2.4 The trajectory of a photon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.2.5 Gravitational waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.2.6 Cosmology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

4 Oscillations 184.1 Harmonic oscillations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

4.2 Mechanic oscillations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184.3 Electric oscillations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194.4 Waves in long conductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194.5 Coupled conductors and transformers . . . . . . . . . . . . . . . . . . . . . . . . . . . 194.6 Pendulums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

5 Waves 205.1 The wave equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205.2 Solutions of the wave equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

5.2.1 Plane waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205.2.2 Spherical waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215.2.3 Cylindrical waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215.2.4 The general solution in one dimension . . . . . . . . . . . . . . . . . . . . . . . 21

5.3 The stationary phase method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215.4 Green functions for the initial-value problem . . . . . . . . . . . . . . . . . . . . . . . 225.5 Waveguides and resonating cavities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225.6 Non-linear wave equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

6 Optics 246.1 The bending of light . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246.2 Paraxial geometrical optics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

6.2.1 Lenses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246.2.2 Mirrors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256.2.3 Principal planes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256.2.4 Magnification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

6.3 Matrix methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256.4 Aberrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266.5 Reflection and transmission . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266.6 Polarization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276.7 Prisms and dispersion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276.8 Diffraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286.9 Special optical effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286.10 The Fabry-Perot interferometer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

7 Statistical physics 307.1 Degrees of freedom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307.2 The energy distribution function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

7.3 Pressure on a wall . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317.4 The equation of state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317.5 Collisions between molecules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32


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7.6 Interaction between molecules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

8 Thermodynamics 33

8.1 Mathematical introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 338.2 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 338.3 Thermal heat capacity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 338.4 The laws of thermodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 348.5 State functions and Maxwell relations . . . . . . . . . . . . . . . . . . . . . . . . . . . 348.6 Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 358.7 Maximal work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 368.8 Phase transitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 368.9 Thermodynamic potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 378.10 Ideal mixtures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 378.11 Conditions for equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 378.12 Statistical basis for thermodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

8.13 Application on other systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

9 Transport phenomena 399.1 Mathematical introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 399.2 Conservation laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 399.3 Bernoulli’s equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 419.4 Caracterising of flows with dimensionless numbers . . . . . . . . . . . . . . . . . . . . 419.5 Tube flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 429.6 Potential theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 429.7 Boundary layers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

9.7.1 Flow boundary layers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 439.7.2 Temperature boundary layers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

9.8 Heat conductance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 449.9 Turbulence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 449.10 Self organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

10 Quantum physics 4510.1 Introduction in quantum physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

10.1.1 Black body radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4510.1.2 The Compton effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4510.1.3 Electron diffraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

10.2 Wave functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4510.3 Operators in quantum physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4510.4 The uncertaincy principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4610.5 The Schrödinger equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

10.6 Parity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4610.7 The tunnel effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4710.8 The harmonic oscillator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4710.9 Angular momentum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4710.10 Spin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4810.11 The Dirac formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4810.12 Atom physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

10.12.1 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4910.12.2 Eigenvalue equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4910.12.3 Spin-orbit interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4910.12.4 Selection rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

10.13 Interaction with electromagnetic fields . . . . . . . . . . . . . . . . . . . . . . . . . . 50

10.14 Perturbation theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5010.14.1 Time-independent perturbation theory . . . . . . . . . . . . . . . . . . . . . . 5010.14.2 Time-dependent perturbation theory . . . . . . . . . . . . . . . . . . . . . . . 51


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10.15 N-particle systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

10.15.1 General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

10.15.2 Molecules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5210.16 Quantum statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

11 Plasma physics 54

11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

11.2 Transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

11.3 Elastic collisions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

11.3.1 General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

11.3.2 The Coulomb interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

11.3.3 The induced dipole interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

11.3.4 The center of mass system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

11.3.5 Scattering of light at free electrons . . . . . . . . . . . . . . . . . . . . . . . . . 56

11.4 Thermodynamic equilibrium and reversibility . . . . . . . . . . . . . . . . . . . . . . . 57

11.5 Inelastic collisions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

11.5.1 Types of collisions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

11.5.2 Cross sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

11.6 Radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

11.7 The Boltzmann transport equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

11.8 Collision-radiative models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

11.9 Waves in plasma’s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

12 Solid state physics 62

12.1 Crystal structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

12.2 Crystal binding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

12.3 Crystal vibrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

12.3.1 lattice with one kind of atoms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

12.3.2 A lattice with two kinds of atoms . . . . . . . . . . . . . . . . . . . . . . . . . . 63

12.3.3 Phonons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

12.3.4 Thermal heat capacity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

12.4 Magnetic field in the solid state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

12.4.1 Dielectrics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

12.4.2 Paramagnetism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

12.4.3 Ferromagnetism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

12.5 Free electron Fermi gas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

12.5.1 Thermal heat capacity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

12.5.2 Electric conductance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

12.5.3 The Hall-effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

12.5.4 Thermal heat conductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

12.6 Energy bands . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

12.7 Semiconductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

12.8 Superconductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

12.8.1 Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

12.8.2 The Josephson effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

12.8.3 Fluxquantisation in a superconducting ring . . . . . . . . . . . . . . . . . . . . 69

12.8.4 Macroscopic quantum interference . . . . . . . . . . . . . . . . . . . . . . . . . 69

12.8.5 The London equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

12.8.6 The BCS model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70


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13 Theory of groups 7113.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

13.1.1 Definition of a group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

13.1.2 The Cayley table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7113.1.3 Conjugated elements, subgroups and classes . . . . . . . . . . . . . . . . . . . . 7113.1.4 Isomorfism and homomorfism; representations . . . . . . . . . . . . . . . . . . . 7213.1.5 Reducible and irreducible representations . . . . . . . . . . . . . . . . . . . . . 72

13.2 The fundamental orthogonality theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 7213.2.1 Schur’s lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7213.2.2 The fundamental orthogonality theorem . . . . . . . . . . . . . . . . . . . . . . 7213.2.3 Character . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

13.3 The relation with quantummechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7313.3.1 Representations, energy levels and degeneracy . . . . . . . . . . . . . . . . . . . 7313.3.2 Breaking of degeneracy with a perturbation . . . . . . . . . . . . . . . . . . . . 7313.3.3 The construction of a basefunction . . . . . . . . . . . . . . . . . . . . . . . . . 73

13.3.4 The direct product of representations . . . . . . . . . . . . . . . . . . . . . . . 7413.3.5 Clebsch-Gordan coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7413.3.6 Symmetric transformations of operators, irreducible tensor operators . . . . . . 7413.3.7 The Wigner-Eckart theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

13.4 Continuous groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7513.4.1 The 3-dimensional translation group . . . . . . . . . . . . . . . . . . . . . . . . 7513.4.2 The 3-dimensional rotation group . . . . . . . . . . . . . . . . . . . . . . . . . . 7613.4.3 Properties of continuous groups . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

13.5 The group SO(3) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7713.6 Applications in quantum mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

13.6.1 Vectormodel for the addition of angular momentum . . . . . . . . . . . . . . . 7813.6.2 Irreducible tensoroperators, matrixelements and selection rules . . . . . . . . . 78

13.7 Applications in particle physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

14 Nuclear physics 8114.1 Nuclear forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8114.2 The shape of the nucleus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8214.3 Radioactive decay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8214.4 Scattering and nuclear reactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

14.4.1 Kinetic model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8314.4.2 Quantummechanical model for n-p scattering . . . . . . . . . . . . . . . . . . . 8314.4.3 Conservation of energy and momentum in nuclear reactions . . . . . . . . . . . 84

14.5 Radiation dosimetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

15 Quantum field theory & Particle physics 85

15.1 Creation and annihilation operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8515.2 Classical and quantum fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8515.3 The interaction picture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8615.4 Real scalar field in the interaction picture . . . . . . . . . . . . . . . . . . . . . . . . 8615.5 Charged spin-0 particles, conservation of charge . . . . . . . . . . . . . . . . . . . . . 8715.6 Field functions for spin- 1

2 particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

15.7 Quantization of spin- 12 fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8815.8 Quantization of the electromagnetic field . . . . . . . . . . . . . . . . . . . . . . . . . 8915.9 Interacting fields and the S-matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8915.10 Divergences and renormalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9015.11 Classification of elementary particles . . . . . . . . . . . . . . . . . . . . . . . . . . . 9115.12 P and CP-violation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

15.13 The standard model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9315.13.1 The electroweak theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9315.13.2 Spontaneous symmetry breaking . . . . . . . . . . . . . . . . . . . . . . . . . 94


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VI Equations in Physics by ir. J.C.A. Wevers

15.13.3 Quantumchromodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9415.14 Pathintegrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

16 Astrophysics 9616.1 Determination of distances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9616.2 Brightnes and magnitudes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9616.3 Radiation and stellar atmospheres . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9716.4 Composition and evolution of stars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9716.5 Energy production in stars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98


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Chapter 1

Mechanics

1.1 Point-kinetics in a fixed coordinate system

1.1.1 Definitions

The position r, the velocity v and the acceleration a are defined by: r = (x,y,z), v = (ẋ, ẏ, ż),a = (ẍ, ÿ, z̈). The following holds:

s(t) = s0 + |v(t)|dt ; r(t) = r0 + v(t)dt ; v(t) = v0 + a(t)dt

When the acceleration is constant this gives: v(t) = v0 + at and s(t) = s0 + v0t + 12 at

2.For the unit vectors in a direction ⊥ to the orbit et and parallel to it en holds:

et = v

|v| = dr

dṡet =

v

ρen ; en =

̇et

|̇et|For the curvature k and the radius of curvature ρ holds:

k = det

ds =

d2r

ds2 =

dϕ

ds

; ρ = 1

|k

|1.1.2 Polar coordinates

Polar coordinates are defined by: x = r cos(θ), y = r sin(θ). So, for the unit coordinate vectors holds:

̇er = θ̇eθ, ̇eθ = −θ̇erThe velocity and the acceleration are derived from: r = rer , v = ṙer + rθ̇eθ, a = (r̈ − rθ̇2)er + ( 2ṙθ̇ +rθ̈)eθ.

1.2 Relative motion

For the motion of a point D w.r.t. a point Q holds: rD = rQ + ω × vQ

ω2 with QD = rD −rQ and ω = θ̇.

Further holds: α = θ̈. ′ means that the quantity is defined in a moving system of coordinates. In amoving system holds:v = vQ + v

′ + ω × r ′ and a = aQ + a ′ + α × r ′ + 2 ω × v − ω × ( ω × r ′)with | ω × ( ω × r ′)| = ω2rn ′

1.3 Point-dynamics in a fixed coordinate system

1.3.1 Force, (angular)momentum and energy

Newton’s 2nd law connects the force on an object and the resulting acceleration of the object:

F (r, v, t) = ma =

d p

dt , where the momentum

is given by: p = mv

Newton’s 3rd law is: F action = − F reaction.

2


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Chapter 1: Mechanics 3

For the power P holds: P = Ẇ = F · v. For the total energy W , the kinetic energy T and thepotential energy U holds: W = T + U ; Ṫ = −U̇ with T = 12 mv2.

The kick S is given by: S = ∆ p = F dtThe work A, delivered by a force, is A =

21

F · ds = 2

1

F cos(α)ds

The torque τ is related to the angular momentum L: τ = ̇ L = r × F ; and

L = r × p = mv × r, | L| = mr2ω. The following holds:

τ = −∂U ∂θ

So, the conditions for a mechanical equilibrium are:

F i = 0 and

τ i = 0.

The force of friction is usually proportional with the force perpendicular to the surface, except whenthe motion starts, when a threshold has to be overcome: F fric = f · F norm · et.

1.3.2 Conservative force fields

A conservative force can be written as the gradient of a potential: F cons = − ∇U . From this followsthat rot F = 0. For such a force field also holds:

F · ds = 0 ⇒ U = U 0 −

r1 r0

F · ds

So the work delivered by a conservative force field depends not on the followed trajectory but only

on the starting and ending points of the motion.

1.3.3 Gravitation

The Newtonian law of gravitation is (in GRT one also uses κ instead of G):

F g = −G m1m2r2

er

The gravitationpotential is then given by V = −Gm/r. From Gauss law then follows: ∇2V = 4πG̺

.

1.3.4 Orbital equations

From the equations of Lagrange for φ, conservation of angular momentum can be derived:

∂ L∂φ

= ∂V

∂φ = 0 ⇒ d

dt(mr2φ) = 0 ⇒ Lz = mr2φ = constant

For the radius as a function of time can be found that:dr

dt

2=

2(W − V )m

− L2

m2r2

The angular equation is then:

φ − φ0 =r

0 mr2

L 2(W − V )

m − L

2

m2r2 −1

dr r−2field

= arccos1 +1r − 1

r01r0

+ km/L2zif F = F (r): L =constant, if F is conservative: W =constant, if F ⊥ v then ∆T = 0 and U = 0.


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4 Equations in Physics by ir. J.C.A. Wevers

Kepler’s equations

In a force field F = kr−2, the orbits are conic sections (Kepler’s 1st law). The equation of the orbit

is:r(θ) =

ℓ

1 + ε cos(θ − θ0) , or: x2 + y2 = (ℓ − εx)2

with

ℓ = L2

Gµ2M tot; ε2 = 1 +

2W L2

G2µ3M 2tot= 1 − ℓ

a ; a =

ℓ

1 − ε2 = k

2W

a is half the length of the long axis of the elliptical orbit in case the orbit is closed. Half the lengthof the short axis is b =

√ aℓ. ε is the excentricity of the orbit. Orbits with an equal ε are equally

shaped. Now, 5 kinds of orbits are possible:

1. k


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2. Rotation: F α = −m α × r ′

3. Coriolis force: F cor =

−2m ω

×v

4. Centrifugal force: F cf = mω2rn ′ = − F cp ; F cp = −mv

2

r er

1.4.2 Tensor notation

Transformation of the Newtonian equations of motion to xα = xα(x) gives:

dxα

dt =

∂xα

∂ ̄xβdx̄β

dt ;

sod

dt

dxα

dt

= d2xα

dt2

= d

dt ∂xα

∂ ̄xβ

dx̄β

dt = ∂xα

∂ ̄xβ

d2x̄β

dt2

+ dx̄β

dt

d

dt ∂xα

∂ ̄xβ

The chain rule gives:d

dt

∂xα

∂ ̄xβ =

∂

∂ ̄xγ ∂xα

∂ ̄xβdx̄γ

dt =

∂ 2xα

∂ ̄xβ∂ ̄xγ dx̄γ

dt

So:d2xα

dt2 =

∂xα

∂ ̄xβd2x̄β

dt2 +

∂ 2xα

∂ ̄xβ∂ ̄xγ dx̄γ

dt

So the Newtonian equation of motion

md2xα

dt2 = F α

will be transformed into:

md2xα

dt2 + Γαβγ dx

β

dtdxγ dt

= F α

The apparent forces are brought from he origin to the effect side in the way Γαβγ dxβ

dt

dxγ

dt .

1.5 Dynamics of masspoint collections

1.5.1 The center of mass

The velocity w.r.t. the center of mass R is given by v − ̇ R. The coordinates of the center of mass aregiven by:

rm = mirimiIn a 2-particle system, the coordinates of the center of mass are given by:

R = m1r1 + m2r2

m1 + m2

With r = r1 −r2, the kinetic energy becomes: T = 12 M tot Ṙ2 + 12 µṙ2, with the reduced mass µ is givenby:

1

µ =

1

m1+

1

m2The motion within and outside the center of mass can be separated:

̇ Loutside = τ outside ;

̇ Linside = τ inside

p = mvm ; F ext = mam ; F 12 = µu


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1.5.2 Collisions

With collisions, where B are the coordinates of the collision and C an arbitrary other position, holds:

p = mvm is constant, and T = 1

2 mv 2m is constant. The changes in the relative velocities can be

derived from: S = ∆ p = µ(vaft − vbefore). Further holds ∆ LC = CB × S , p S =constant and Lw.r.t. B is constant.

1.6 Dynamics of rigid bodies

1.6.1 Moment of Inertia

The angular momentum in a moving coordinate system is given by:

L′ = I ω + L′n

where I is the moment of inertia with respect to a central axis, which is given by:

I =i

miri2 ; T ′ = W rot =

12

ωI ijeiej = 12

Iω2

or, in the continuous case:

I = m

V

r′ndV =

r′ndm

Further holds:

Li = I ijωj ; I ii = I i ; I ij = I ji = −

k

mkx′ix′j

Steiner’s theorem is: I w.r.t.D = I w.r.t.C + m(DM )2 if axis C

axis D.

Object I Object I

Cavern cylinder I = mR2 Massive cylinder I = 12

mR2

Disc, axis in plane disc through m I = 14

mR2 Halter I = 12

µR2

Cavern sphere I = 23 mR2 Massive sphere I = 25 mR

2

Bar, axis ⊥ through c.o.m. I = 112

ml2 Bar, axis ⊥ through end I = 13

ml2

Rectangle, axis ⊥ plane thr. c.o.m. I = 112 (a2 + b2) Rectangle, axis b thr. m I = ma2

1.6.2 Principal axes

Each rigid body has (at least) 3 principal axes which stand ⊥ at each other. For a principal axisholds:

∂I

∂ωx=

∂I

∂ωy=

∂I

∂ωz= 0 so L′n = 0

The following holds: ω̇k = −aijkωiωj with aijk = I i − I jI k

if I 1 ≤ I 2 ≤ I 3.

1.6.3 Time dependence

For torque of force τ holds:

τ ′

= I ¨θ ;

d′′ L′

dt = τ ′

− ω × L′

The torque T is defined by: T = F × d.


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1.7 Variational Calculus, Hamilton and Lagrange mechanics

1.7.1 Variational Calculus

Starting with:

δ

b a

L(q, q̇, t)dt = 0 met δ (a) = δ (b) = 0 and δ

du

dx

=

d

dx(δu)

the equations of Lagrange can be derived:

d

dt

∂ L∂ q̇ i

= ∂ L∂q i

When there are additional conditions applying on the variational problem δJ (u) = 0 of the type

K (u) =constant, the new problem becomes: δJ (u) − λδK (u) = 0.

1.7.2 Hamilton mechanics

The Lagrangian is given by: L = T (q̇ i) − V (q i). The Hamiltonian is given by: H = q̇ i pi −L. In2 dimensions holds: L = T − U = 1

2m(ṙ2 + r2 φ̇2) − U (r, φ).

If the used coordinates are canonical are the Hamilton equations the equations of motion for thesystem:

dq idt

= ∂H

∂pi;

dpidt

= −∂H ∂q i

Coordinates are canonical if the following holds: {q i, q j} = 0, { pi, pj} = 0, {q i, pj} = δ ij where {, }is the Poisson bracket : {A, B} =

i

∂A∂q i

∂B

∂pi− ∂A

∂pi

∂B

∂q i

1.7.3 Motion around an equilibrium, linearization

For natural systems around equilibrium holds:∂V

∂q i

0

= 0 ; V (q ) = V (0) + V ikq iq k with V ik =

∂ 2V

∂q i∂q k

0

With T = 12

(M ik q̇ i q̇ k) one receives the set of equations M q̈ + V q = 0. If we substitute q i(t) =ai exp(iωt), this set of equations has solutions if det(V

−ω2M ) = 0. This leads to the eigenfrequentions

of the problem: ω2k = aTk V akaTk M ak

. If the equilibrium is stable holds: ∀k that ω2k > 0. The general solutionis a superposition if eigenvibrations.

1.7.4 Phase space, Liouville’s equation

In phase space holds:

∇ =

i

∂

∂q i,i

∂

∂pi

so ∇ · v =

i

∂

∂q i

∂H

∂pi− ∂

∂pi

∂H

∂q i

If the equation of continuity, ∂ t̺ +

∇ ·(̺v ) = 0 holds, this can be written as:

{̺, H } + ∂̺∂t

= 0


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For an arbitrary quantity A holds:dA

dt = {A, H } + ∂ A

∂t

Liouville’s theorem can than be written as:

d̺dt

= 0 ; or:

pdq = constant

1.7.5 Generating functions

Starting with the coordinate transformation: Qi = Qi(q i, pi, t)P i = P i(q i, q i, t)

The following Hamilton equations can be derived:

dQidt

= ∂K

∂P i;

dP idt

= − ∂K ∂Qi

Now, a distinction between 4 cases can be made:

1. If pi q̇ i − H = P iQi − K (P i, Qi, t) − dF 1(q i, Qi, t)dt

, the coordinates follow from:

pi = ∂F 1

∂q i; P i =

∂F 1∂Qi

; K = H + dF 1

dt

2. If pi q̇ i − H = − Ṗ iQi − K + dF 2(q i, P i, t)dt


pi = ∂F 2

∂q i; Qi =

∂F 2∂P i

; K = H + ∂F 2

∂t

3. If − ˙ piq i − H = P i Q̇i − K + dF 3( pi, Qi, t)dt


q i = −∂F 3∂pi

; P i = −∂F 3∂Qi

; K = H + ∂ F 3

∂t

4. If − ˙ piq i − H = −P iQi − K + dF 4( pi, P i, t)dt


q i = −∂F 4∂pi

; Qi = ∂F 4

∂pi; K = H + ∂F 4

∂t

The functions F 1, F 2, F 3 and F 4 are called generating functions .

The Hamiltonian of a charged particle with charge q in an external electromagnetic field is given by:

H = 1

2m

p − q A

2+ qV

This Hamiltonian can be derived from the Hamiltonian of a free particle H = p2/2m with the

transformations p → p−q A and H → H −qV . This is elegant from a relativistic point of view: this isequivalent with the transformation of the momentum 4-vector pα → pα−qAα. A gaugetransformationon the potentials A

α

corresponds with a canonical transformation, which make the Hamilton equationsthe equations of motion for the system.


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Chapter 2

Electricity & Magnetism

2.1 The Maxwell equations

The classical electromagnetic field can be described with the Maxwell equations , and can be writtenboth as differential and integral equations:

( D

·n)d2A = Qfree,included

∇ · D = ρfree

( B · n)d2A = 0 ∇ · B = 0 E · ds = −dΦ

dt ∇ × E = −∂

B

∂t H · ds = I free,included + dΨ

dt ∇ × H = J free + ∂

D

∂t

For the fluxes holds: Ψ =

( D · n)d2A, Φ =

( B · n)d2A.

The electric displacement D, polarization P and electric field strength E depend on each otheraccording to:

D = ε0 E + P = ε0εr E , P = p0/Vol, εr = 1 + χe, with χe = np20

3ε0kT

The magnetic field strength H , the magnetization M and the magnetic flux density B depend oneach other according to:

B = µ0( H + M ) = µ0µr H , M =

m/Vol, µr = 1 + χm, with χm = µ0nm20

3kT

2.2 Force and potential

The force and the electric field between 2 point charges are given by:

F 12 = Q1Q24πε0εrr2 er ; E =

F

Q

The Lorentzforce is the force which is felt by a charged particle that moves through a magnetic field.The origin of this force is a relativistic transformation of the Coulomb force: F L = Q(v× B) = l( I × B).The magnetic field which results from an electric current is given by the law of Biot-Savart:

d B = µ0I

4πr2d l × er

If the current is time-dependent one has to take retardation into account: the substitution I (t) →I (t − r/c) has to be applied.The potentials are given by:

V 12 = −2

1

E · ds , A = 12 B × r

9


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Here, the freedom remains to apply a gauge transformation . The fields can be derived from thepotentials as follows:

E = −∇V − ∂ A

∂t , B = ∇ × AFurther holds the relation: c2 B = v × E .

2.3 Gauge transformations

The potentials of the electromagnetic fields transform as follows when a gauge transformation isapplied:

A′ = A − ∇f V ′ = V +

∂ f

∂t

so the fields E and B do not change. This results in a canonical transformation of the Hamiltonian.Further, the freedom remains to apply a limiting condition. Two common choices are:

1. Lorentz-gauge: ∇ · A + 1c2

∂V

∂t = 0. This separates the differential equations for A and V :

V = − ρε0

, A = −µ0 J .

2. Coulomb gauge: ∇ · A = 0. If ρ = 0 and J = 0 holds V = 0 and follows A from A = 0.

2.4 Energy of the electromagnetic field

The energy density of the electromagnetic field is:

dW dVol

= w = HdB + EdDThe energy density can be expressed in the potentials and currents as follows:

wmag = 12

J · Ad3x , wel = 12

ρV d3x

2.5 Electromagnetic waves

2.5.1 Electromagnetic waves in vacuum

The wave equation Ψ(r, t) = −f (r, t) has the general solution, with c = (ε0µ0)−1/2:

Ψ(r, t) = f (r, t − |r − r ′|/c)

4π|r − r ′| d3r′

If this is written as: J (r, t) = J (r)exp(−iωt) and A(r, t) = A(r)exp(−iωt) with:

A(r) = µ

4π

J (r ′)

exp(ik|r − r ′|)|r − r ′| d

3r ′ , V (r) = 1

4πε

ρ(r ′)

exp(ik|r − r ′|)|r − r ′| d

3r ′

will a derivation via multipole development show that for the radiated energy holds, if d, λ ≫ r:dP

dΩ =

k2

32π2ε0c

J ⊥(r

′)ei k·rd3r′

2

The energy density of the electromagnetic wave of a vibrating dipole at a large distance is:

w = ε0E 2 =

p20 sin2(θ)ω4

16π2ε0r2c4 sin2(kr − ωt) , wt =

p20 sin2(θ)ω4

32π2ε0r2c4 , P =

ck4| p |212πε0


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Chapter 2: Electricity & Magnetism 11

The radiated energy can be derived from the Poynting vector S : S = E × H = cWev. The irradiance is the time-averaged of the Poynting vector: I = | S |t. The radiation pressure ps is given by ps = (1 + R)

| S |/c, where R is the coefficient of reflection.

2.5.2 Electromagnetic waves in mater

The wave equations in matter, with cmat = (εµ)−1/2 are:

∇2 − εµ ∂ 2

∂t2 − µ

ρ

∂

∂t

E = 0 ,

∇2 − εµ ∂

2

∂t2 − µ

ρ

∂

∂t

B = 0

give, after substitution of monochromatic plane waves: E = E exp(i( k · r − ωt)) and B = B exp(i( k ·r − ωt)) the dispersion relation:

k2 = εµω2 + iµω

ρ

The first term arises from the displacement current, the second from the conductance current. If k iswritten as k := k′ + ik′′ holds:

k′ = ω

12

εµ

1 +

1 + 1

(ρεω)2 and k′′ = ω

12

εµ

−1 +

1 + 1

(ρεω)2

This results in a damped wave: E = E exp(−k′′n · r)exp(i(k′n · r − ωt)). If the material is a goodconductor, the wave vanishes after approximately one wavelength, k = (1 + i)

µω

2ρ .

2.6 Multipoles

Because 1

|r − r ′| = 1

r

∞0

r′r

lP l(cos θ) can the potential be written as: V =

Q

4πε

n

knrn

For the lowest-order terms this results in:

• Monopole: l = 0, k0 =

ρdV

• Dipole: l = 1, k1 =

r cos(θ)ρdV

• Quadrupole: l = 2, k2 = 12i

(3z2i − r2i )

1. The electric dipole: dipolemoment: p = Qle, where e goes from ⊕ to ⊖, and F = ( p · ∇) E ext,and W =

− p

· E out.

Electric field: E ≈ Q4πεr3

3 p · rr2

− p. The torque is: τ = p × E out2. The magnetic dipole: dipolemoment: if r ≫ √ A: µ = I × (Ae⊥), F = ( µ · ∇) Bout

|µ| = mv2⊥

2B , W = − µ × Bout

Magnetic field: B = −µ4πr3

3µ · r

r2 − µ

. The moment is: τ = µ × Bout

2.7 Electric currents

The continuity equation for charge is: ∂ρ

∂t + ∇ · J = 0. The electric current is given by:

I = dQ

dt =

( J · n)d2A


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For most conductors holds: J = E/ρ, where ρ is the resistivity .

If the flux enclosed by a conductor changes this results in an induction voltage V ind =

−N

dΦ

dt

. If the

current flowing through a conductor changes, this results in a self-inductance voltage which works

against the change: V selfind = −L dI dt

. If a conductor encloses a flux Φ holds: Φ = LI .

The magnetic induction within a coil is approximated by: B = µN I √

l2 + 4R2 where l is the length, R

the radius and N the number of coils. The energy contained within a coil is given by W = 12 LI 2 and

L = µN 2A/l.

The Capacity is defined by: C = Q/V . For a capacitor holds: C = ε0εrA/d where d is the distancebetween the plates and A the surface of one plate. The electric field strength between the plates isE = σ/ε0 = Q/ε0A where σ is the surface charge. The accumulated energy is given by W =

12

CV 2.

The current through a capacity is given by I =

−C

dV

dt

.

For most PTC resistors holds approximately: R = R0(1 + αT ), where R0 = ρl/A. For a NTC holds:R(T ) = C exp(−B/T ) where B and C depend only on the material.If a current flows through two different, connecting conductors x and y , the contact area will heat upor cool down, depending on the direction of the current: the Peltier effect . The generated or removedheat is given by: W = ΠxyIt. This effect can be amplified with semiconductors.

The thermic voltage between 2 metals is given by: V = γ (T −T 0). For a Cu-Konstantane connectionholds: γ ≈ 0, 2 − 0.7 mV/K.In an electrical net with only stationary currents, Kirchhoff’s equations apply: for a knot holds:

I n = 0, along a closed path holds:

V n =

I nRn = 0.

2.8 Depolarizing field

If a dielectric material is placed in an electric or magnetic field, the field strength within and outsidethe material will change because the material will be polarized or magnetized. If the medium hasan ellipsoidal shape and one of the principal axes are parallel with the external field E 0 or B0 is thedepolarizing field homogeneous.

E dep = E mat − E 0 = − N P

ε0 H dep = H mat − H 0 = −N M

N is a constant depending only on the shape of the object placed in the field, with 0

≤ N ≤1. For

a few limiting cases of an ellipsoid holds: a thin plane: N = 1, a long, thin bar: N = 0, a sphere: N = 13 .

2.9 Mixtures of materials

The average electric displacement in a material which in inhomogenious on a mesoscopic scale is given

by: D = εE = ε∗ E where ε∗ = ε1

1 − φ2(1 − x)Φ(ε∗/ε2)

−1where x = ε1/ε2. For a sphere holds:

Φ = 13

+ 23

x. Further holds:

iφiεi

−1

≤ ε∗ ≤ iφiεi


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Chapter 3

Relativity

3.1 Special relativity

3.1.1 The Lorentz transformation

The Lorentz transformation ( x′, t′) = (x′(x, t), t′( x, t)) leaves the wave equation invariant if c isinvariant:

∂ 2

∂x2 + ∂ 2

∂y 2 + ∂ 2

∂z 2 − 1

c2∂ 2

∂t2 = ∂ 2

∂x′2 + ∂ 2

∂y ′2 + ∂ 2

∂z ′2 − 1

c2∂ 2

∂t′2

This transformation can also be found when ds2 = ds′2 is demanded. The general form of the Lorentztransformation is given by:

x′ = x + (γ − 1)(x · v)v

|v|2 − γvt , t′ =

γ (t − x · v)c2

where

γ = 1

1 − v2c2

If the velocity is parallel to the x-axis, this becomes:

x′ = γ (x − vt) x = γ (x′ + vt′)y′ = y z′ = z

t′ = γ

t − xvc2

t = γ

t′ +

x′v

c2

The velocity difference v ′ between two observers transforms according to:

v ′ =

γ

1 − v1 · v2

c2

−1 v2 + (γ − 1)v1 · v2

v22v1 − γv1

If v = vex holds:

p′

x = γ px − βW c , W ′ = γ (W − vpx)With β = v/c the electric field of a moving charge is given by:

E = Q

4πε0r2(1 − β 2)er

(1 − β 2 sin2(θ))3/2

The electromagnetic field transforms according to:

E ′ = γ ( E + v × B)′ , B′ = γ

B − v × E

c2

Length, mass and time transform according to: ∆tb = γ ∆te, mb = γm0, lb = l0/γ , with e the

quantities in the frame of he observer and b the quantities outside it. The proper time τ is defined as:dτ 2 = ds2/c2, so ∆τ = ∆t/γ . For energy and momentum holds: W = mc2 = γ W 0, W 2 = m20c

4 + p2c2. p = mv = W v/c2, and pc = W β where β = v/c.

13


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4-vectors have the property that their modulus is independent of the observer: their componentscan change after a coordinate transformation but not their modulus. The difference of two 4-vectors

transforms also as a 4-vector. The 4-vector for the velocity is given by U

α

=

dxα

dτ . The relation withthe “common” velocity ui := dxi/dt is: U α = (γui,icγ ). For particles with nonzero restmass holds:U αU α = −c2, for particles with zero restmass (so with v = c) holds: U αU α = 0. The 4-vector forenergy and momentum is given by: pα = m0U

α = (γpi,iW/c). So: pα pα = −m20c2 = p2 − W 2/c2.

3.1.2 Red and blue shift

There are three causes of red and blue shifts:

1. Motion: with ev · er = cos(ϕ) follows: f ′

f = γ

1 − v cos(ϕ)

c

.

This can give both red- and blueshift, also ⊥ the direction of motion.

2. Gravitational redshift:

∆f

f =

κM

rc2 .

3. Redshift because the universe expands, resulting in e.g. the cosmic background radiation: λ0λ1

=

R0R1

.

3.1.3 The stress-energy tensor and the field tensor

The stress-energy tensor is given by:

T µν = (̺c2 + p)uµuν + pgµν +

1

c2 F µαF

αν + 14 gµν F αβF αβ

The conservation laws can than be written as: ∇ν T µν = 0. The electromagnetic field tensor is givenby:

F αβ = ∂Aβ∂xα

− ∂Aα∂xβ

with Aµ := ( A,iV/c) and J µ := ( J,icρ). The Maxwell equations can than be written as:

∂ ν F µν = µ0J

µ , ∂ λF µν + ∂ µF νλ + ∂ ν F λµ = 0

The equations of motion for a charged particle in an EM field become with the field tensor:

dpαdτ

= qF αβuβ

3.2 General relativity

3.2.1 Riemannian geometry, the Einstein tensor

The basic principles of general relativity are:

1. The geodesic postulate: free falling particles move along geodesics of space-time with the propertime τ or arc length s as parameter. For particles with zero rest mass (photons), the use of afree parameter is required because for them holds ds = 0. From δ

ds = 0 the equations of

motion can be derived:d2xα

ds2 + Γαβγ

dxβ

ds

dxγ

ds = 0

2. The principle of equivalence : inertial mass ≡ gravitational mass ⇒ gravitation is equivalentwith a curved space-time were particles move along geodesics.


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Chapter 3: Relativity 15

3. By a proper choice of the coordinate system it is possible to make the metric locally flat in eachpoint xi: gαβ(xi) = ηαβ :=diag(−1, 1, 1, 1).

The Riemann tensor is defined as: RµναβT ν := ∇α∇βT µ − ∇β∇αT µ, where the covariant derivate is

given by ∇jai = ∂ jai + Γijkak and ∇jai = ∂ jai − Γkijak. Here,

Γijk = ∂ 2x̄l

∂xj∂xk∂xi

∂ ̄xl

are the Christoffel symbols . For a second-order tensor holds: [∇α, ∇β ]T µν = RµσαβT σν + RσναβT µσ ,∇kaij = ∂ kaij − Γlkjail + Γiklalj , ∇kaij = ∂ kaij − Γlkialj − Γlkjajl and ∇kaij = ∂ kaij + Γiklalj + Γjklail.The following holds: Rαβµν = ∂ µΓ

αβν − ∂ ν Γαβµ + ΓασµΓσβν − Γασν Γσβµ .

The Ricci tensor is a contraction of the Riemann tensor: Rαβ := Rµαµβ, which is symmetric: Rαβ =

Rβα . The Bianchi identities are: ∇λRαβµν + ∇ν Rαβλµ + ∇µRαβνλ = 0.The Einstein tensor is given by: Gαβ := Rαβ − 12 gαβR, where R := Rαα is the Ricci scalar , for whichholds: ∇βGαβ = 0. With the variational principle δ

(L(gµν ) − Rc2/16πκ)

|g|d4x = 0 for variationsgµν → gµν + δgµν the Einstein field equations can be derived:

Gαβ = 8πκ

c2 T αβ , which can also be written as Rαβ =

8πκ

c2 (T αβ − 12 gαβT µµ )

For empty space this is equivalent with Rαβ = 0. The equation Rαβµν = 0 has as only solution a flatspace.

The Einstein equations are of 10 independent equations, who are second order in gµν . From here,the Laplace equation from Newtonian gravitation can be derived by stating: gµν = ηµν + hµν , where

|h| ≪ 1. In the stationary case, this results in ∇2

h00 = 8πκ̺

/c

2

.

The most general shape of the field equations is: Rαβ − 12 gαβR + Λgαβ = 8πκ

c2 T αβ

where Λ is the cosmological constant . This constant plays a role in inflatory models of the universe.

3.2.2 The line element

The metric tensor is given by: gij =k

∂ ̄xk

∂xi∂ ̄xk

∂xj .

In general holds: ds2 = gµν dxµdxν . In special relativity this becomes ds2 = −c2dt2 + dx2 + dy2 + dz2.This metric, ηµν :=diag(−1, 1, 1, 1), is called the Minkowski metric .The external Schwarzschild metric applies in vacuum outside a spherical mass distribution, and isgiven by:

ds2 =

−1 + 2m

r

c2dt2 +

1 − 2m

r

−1dr2 + r2dΩ2

Here, m := Mκ/c2 is the geometrical mass of an object with mass M , and dΩ2 = dθ2 + sin2 θdϕ2.This metric is singular for r = 2m = 2κM/c2. If an ob ject is smaller than its eventhorizon 2m it iscalled a black hole . The Newtonian limit of this metric is given by:

ds2 = −(1+ 2V )c2dt2 + (1 − 2V )(dx2 + dy2 + dz2)

where V = −κM/r is the Newtonian gravitation potential. In general relativity, the components of gµν are associated with the potentials and the derivates of gµν with the field strength.

The Kruskal-Szekeres coordinates are used to solve certain problems with the Schwarzschild metricnear r = 2m. They are defined by:


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• r > 2m:

u =

r

2m − 1exp

r

4m cosh t

4mv =

r

2m − 1exp

r4m

sinh

t

4m

• r


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Chapter 3: Relativity 17

3.2.4 The trajectory of a photon

For the trajectory of a photon (and for each particle with zero restmass) holds ds2 = 0. Substituting

the external Schwarzschild metric results in the following orbital equation:

du

dϕ

d2u

dϕ2 + u − 3mu

= 0

3.2.5 Gravitational waves

Starting with the approximation gµν = ηµν + hµν for weak gravitational fields, and the definitionh′µν = hµν − 12 ηµν hαα, follows that h′µν = 0 if the gauge condition ∂ h′µν /∂xν = 0 is satisfied. Fromthis, it follows that the loss of energy of a mechanical system, if the occurring velocities are ≪ c andfor wavelengths ≫ the size of the system, is given by:

dE

dt =

− G

5c5 i,j d3Qij

dt3 2

with Qij =

̺ (xixj − 13 δ ijr2)d3x the mass quadrupole moment.

3.2.6 Cosmology

If for the universe as a whole is assumed:

1. There exists a global time coordinate which acts as x0 of a Gaussian coordinate system,

2. The 3-dimensional spaces are isotrope for a certain value of x0,

3. Each point is equivalent to each other point for a fixed x0.

then the Robertson-Walker metric can be derived for the line element:

ds2 = −c2dt2 + R2(t)

r20

1 − kr

2

4r20

(dr2 + r2dΩ2)

For the scalefactor R(t) the following equations can be derived:

2 R̈

R +

Ṙ2 + kc2

R2 = −8πκp

c2 and

Ṙ2 + kc2

R2 =

8πκ̺

3

where p is the pressure and ̺ the density of the universe. For the deceleration parameter q followsfrom this:

q = − R̈RṘ2

= 4πκ̺3H 2

where H = Ṙ/R is Hubble’s constant . This is a measure of the velocity of which galaxies far awayare moving away of each other, and has the value ≈ (75 ± 25) km·s−1·Mpc−1. This gives 3 possibleconditions of the universe (here, W is the total amount of energy in the universe):

1. Parabolical universe: k = 0, W = 0, q = 12 . The expansion velocity of the universe → 0 if t → ∞. The hereto related density ̺ c = 3H 2/8πκ is the critical density .

2. Hyperbolical universe: k = −1, W 0, q > 12

. The expansion velocity of the universe becomes

negative after some time: the universe starts falling together.


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Chapter 4

Oscillations

4.1 Harmonic oscillations

The general shape of a harmonic oscillation is: Ψ(t) = Ψ̂ei(ωt±ϕ) ≡ Ψ̂cos(ωt ± ϕ),where Ψ̂ is the amplitude . A superposition of more harmonic oscillations with the same frequencyresults in an other harmonic oscillation:

i

Ψ̂i cos(αi ± ωt) = Φ̂cos(β ± ωt)

with:

tan(β ) =

i

Ψ̂i sin(αi)i

Ψ̂i cos(αi)and Φ̂2 =

i

Ψ̂2i + 2j>i

i

Ψ̂iΨ̂j cos(αi − αj)

For harmonic oscillations holds:

x(t)dt =

x(t)

iω and

dnx(t)

dtn = (iω)nx(t).

4.2 Mechanic oscillations

For a construction with a spring with constant C parallel to a damping k which is connected toa mass M , on which a periodic force F (t) = F̂ cos(ωt) is applied holds the equation of motionmẍ = F (t) − kẋ − Cx. With complex amplitudes, this becomes −mω2x = F − Cx − ikωx. Withω20 = C/m follows:

x = F

m(ω20 − ω2) + ikω , and for the velocity holds: ẋ =

F

i√

Cmδ + k

where δ = ω

ω0− ω0

ω . The quantity Z = F /ẋ is called the impedance of the system. The quality of the

system is given by Q =

√ Cmk .

The frequency with minimal |Z | is called velocity resonance frequency . This is equal to ω0. In theresonance curve is |Z |/√ Cm plotted against ω/ω0. The width of this curve is characterized by thepoints where |Z (ω)| = |Z (ω0)|

√ 2. In these points holds: R = X and δ = ±Q−1, and the width is

2∆ωB = ω0/Q.

The stiffness of an oscillating system is given by F/x. The amplitude resonance frequency ωA is the

frequency where iωZ is minimal. This is the case for ωA = ω0

1 − 1

2Q2.

The damping frequency ωD is a measure for the time in which an oscillating system comes to rest. It

is given by ωD = ω0 1 − 1

4Q2

. A weak damped oscillation (k2 4mC )

drops like (if k 2 ≫ 4mC ) x(t) ≈ x0 exp(−t/τ ).

18


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Chapter 4: Oscillations 19

4.3 Electric oscillations

The impedance is given by: Z = R + iX . The phase angle is ϕ := arctan(X/R). The impedance of

a resistor is R, of a capacitor 1/iωC and of a self inductor iωL. The quality of a coil is Q = ωL/R.The total impedance in case several elements are positioned is given by:

1. Series connection: V = I Z ,

Z tot =i

Z i , Ltot =i

Li , 1

C tot=i

1

C i, Q =

Z 0R

, Z = R(1 + iQδ )

2. parallel connection: V = I Z ,

1

Z tot=i

1

Z i,

1

Ltot=i

1

Li, C tot =

i

C i , Q = R

Z 0, Z =

R

1 + iQδ

Here, Z 0 = L

C and ω0 = 1√

LC .

The power given by a source is given by P (t) = V (t) · I (t), soP t = 12 V̂ Î cos(φv − φi) = V̂ eff Î eff cos(∆φ) where cos(∆φ) is the work factor.

4.4 Waves in long conductors

These cables are in use for signal transfer, e.g. coax cable. For them holds: Z 0 =

dL

dx

dx

dC .

The transmission velocity is given by v =

dx

dL

dx

dC .

4.5 Coupled conductors and transformers

For two coils enclosing each others flux holds: if Φ12 is the part of the flux originating from I 2 throughcoil 2 which is enclosed by coil 1, than holds Φ12 = M 12I 2, Φ21 = M 21I 1. For the coefficients of mutualinduction M ij holds:

M 12 = M 21 := M = k

L1L2 = N 1Φ1

I 2=

N 2Φ2I 1

∼ N 1N 2

where 0 ≤ k ≤ 1 is the coupling factor . For a transformer it is k ≈ 1. At full load holds:V 1

V 2=

I 2

I 1=

− iωM

iωL2 + Rload ≈ − L1

L2=

−N 1

N 2

4.6 Pendulums

The oscillation time T = 1/f , and for different types of pendulums is given by:

• Oscillating spring: T = 2π m/C if the spring force is given by F = C · ∆l.• Physical pendulum: T = 2π I/τ with τ the moment of force and I the moment of inertia.• Torsion pendulum: T = 2π I/κ with κ = 2lm

πr4∆ϕ the constant of torsion and I the moment

of inertia.

• Mathematic pendulum: T = 2π l/g with g the acceleration of gravity and l the length of thependulum.


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Chapter 5

Waves

5.1 The wave equation

The general shape of the wave equation is: u = 0, or:

∇2u − 1v2

∂ 2u

∂t2 =

∂ 2u

∂x2 +

∂ 2u

∂y 2 +

∂ 2u

∂z 2 − 1

v2∂ 2u

∂t2 = 0

where u is the disturbance and v the propagation velocity . In general holds: v = f λ. Per definitionholds: kλ = 2π and ω = 2πf .

In principle, there are two kinds of waves:

1. Longitudinal waves: for these holds k v u.2. Transversal waves: for these holds k v ⊥ u.

The phase velocity is given by vph = ω/k. The group velocity is given by:

vg = dω

dk = vph + f

dvphdk

= vph

1 − k

n

dn

dk

where n is the refractive index of the medium. If one want to transfer information with a wave, e.g.by modulating it, will the information move with the group velocity. If vph does not depend on ωholds: vph = vg. In a dispersive medium it is possible that vg > vph or vg < vph.For some media, the propagation velocity follows from:

• Pressure waves in a liquid or gas: v = κ/̺

, where κ is the modulus of compression.

• For pressure waves in a gas also holds: v = γp/̺

=

γRT/M .

• Pressure waves in a solid bar: v = E/̺

• waves in a string: v =

F spanl/m

• Surface waves on a liquid: v = gλ

2π +

2πγ ̺λ

tanh

2πhλ

where h is the depth of the liquid and γ the surface tension. If h ≪ λ holds: v ≈ √ gh.

5.2 Solutions of the wave equation

5.2.1 Plane waves

In n dimensions a harmonic plane wave is defined by:

u(x, t) = 2nû cos(ωt)n

i=1 sin(kixi)The equation for a harmonic traveling plane wave is: u(x, t) = û cos( k · x ± ωt + ϕ)

20


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Chapter 5: Waves 21

If waves reflect at the end of a spring will this result in a change in phase. A fixed end gives a phasechange of π/2 to the reflected wave, with boundary condition u(l) = 0. A lose end gives no changein the phase of the reflected wave, with boundary condition (∂u/∂x)l = 0.

If an observer is moving w.r.t. the wave with a velocity vobs, he will observe a change in frequency:

the Doppler effect . This is given by: f

f 0=

vf − vobsvf

.

5.2.2 Spherical waves

When the situation is spherical symmetric, the homogeneous wave equation is given by:

1

v2∂ 2(ru)

∂t2 − ∂

2(ru)

∂r2 = 0

with general solution:

u(r, t) = C 1f (r − vt)

r + C 2

g(r + vt)

r

5.2.3 Cylindrical waves

When the situation has a cylindrical symmetry, the homogeneous wave equation becomes:

1

v2∂ 2u

∂t2 − 1

r

∂

∂r

r

∂u

∂r

= 0

This is a Bessel equation, with solutions which can be written as Hankel functions. For sufficientlarge values of r these are approximated with:

u(r, t) = û√

r cos(k(r ± vt))

5.2.4 The general solution in one dimensionStarting point is the equation:

∂ 2u(x, t)

∂t2 =

N m=0

bm

∂ m

∂xm

u(x, t)

where bm ∈ IR. Substituting u(x, t) = Aei(kx−ωt) gives two solutions ωj = ωj(k) as dispersionrelations. The general solution is given by:

u(x, t) =

∞ −∞

a(k)ei(kx−ω1(k)t) + b(k)ei(kx−ω2(k)t)

dk

Because in general the frequencies ωj are non-linear in k there is dispersion and the solution can not

be written any more as a sum of functions depending only on x ± vt: the wave front transforms.

5.3 The stationary phase method

The Fourier integrals of the previous section can usually not be calculated exact. If ωj(k) ∈ IR thestationary phase method can be applied. Assuming that a(k) is only a slowly varying function of k ,one can state that the parts of the k-axis where the phase of kx−ω(k)t changes rapidly will give no netcontribution to the integral because the exponent oscillates rapidly there. The only areas contributing

significantly to the integral are areas with a stationary phase, determined by d

dk(kx − ω(k)t) = 0.

Now the following approximation is possible:∞

−∞

a(k)ei(kx−ω(k)t)dk ≈N i=1

2πd2ω(ki)dk2i

exp −i 14 π + i(kix − ω(ki)t)


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5.4 Green functions for the initial-value problem

This method is preferable if the solutions deviate much from the stationary solutions, like point-like

excitations. Starting with the wave equation in one dimension, with ∇2 = ∂ 2/∂x2 holds: if Q(x, x′, t)is the solution with initial values Q(x, x′, 0) = δ (x − x′) and ∂Q(x, x

′, 0)

∂t = 0, and P (x, x′, t) the

solution with initial values P (x, x′, 0) = 0 and ∂P (x, x′, 0)

∂t = δ (x − x′), then the solution of the wave

equation with arbitrary initial conditions f (x) = u(x, 0) and g (x) = ∂u(x, 0)

∂t is given by:

u(x, t) =

∞ −∞

f (x′)Q(x, x′, t)dx′ +

∞ −∞

g(x′)P (x, x′, t)dx′

P and Q are called the propagators . They are defined by: door:

Q(x, x′, t) = 12 [δ (x − x′ − vt) + δ (x − x′ + vt)]

P (x, x′, t) =

1

2v if |x − x′| < vt

0 if |x − x′| > vt

Further holds the relation: Q(x, x′, t) = ∂P (x, x′, t)

∂t

5.5 Waveguides and resonating cavities

The boundary conditions at a perfect conductor can be derived from the Maxwell equations. If n isa unit vector ⊥ the surface, aimed from 1 to 2, and K is a surface current density, than holds:

n · ( D2 − D1) = σ n × ( E 2 − E 1) = 0n · ( B2 − B1) = 0 n × ( H 2 − H 1) = K

In a waveguide holds because of the cylindrical symmetry: E (x, t) = E (x, y)ei(kz−ωt) and B(x, t) = B(x, y)ei(kz−ωt). One can now deduce that, if Bz and E z are not ≡ 0:

Bx = iεµω2 − k2

k

∂ Bz∂x

− εµω ∂ E z∂y

By = i

εµω2 − k2

k∂ Bz∂y

+ εµω∂ E z∂x

E x = i

εµω2 − k2

k∂ E z∂x

+ εµω∂ Bz∂y

E y = i

εµω2 − k2

k∂ E z∂y

− εµω ∂ Bz∂x

Now one can distinguish between three cases:

1. Bz ≡ 0: the Transversal Magnetic modes (TM). Boundary condition: E z|surf = 0.

2. E z ≡ 0: the Transversal Electric modes (TE). Boundary condition: ∂ Bz∂n

surf

= 0.

For the TE and TM modes this gives an eigenvalue problem for E z resp. Bz with as boundaryconditions:

∂ 2

∂x2 +

∂ 2

∂y 2

ψ = −γ 2ψ with eigenvalues γ 2 := εµω2 − k2

This gives a discrete solution ψℓ at eigenvalue γ 2ℓ : k =

εµω2 − γ 2ℓ . For ω < ωℓ, k is imaginaryand the wave is damped. Therefore, ωℓ is called the cut-off frequency . In rectangular conductorsthe following expression can be found for the cut-off frequency for modes TEm,n of TMm,n:

λℓ = 2

(m/a)2 + (n/b)2


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Chapter 5: Waves 23

3. E z and Bz are zero everywhere: the Transversal electromagnetic mode (TEM). Than holds:k = ±ω√ εµ and vf = vg, just as if here were no waveguide. Further k ∈ IR, so there exist nocut-off frequency.

In a rectangular, 3 dimensional resonating cavity with edges a, b and c are the possible wave numbers

given by: kx = n1π

a , ky =

n2π

b , kz =

n3π

c This results in the possible frequencies f = vk/2π in

the cavity:

f = v

2

n2xa2

+n2yb2

+ n2zc2

For a cubic cavity, with a = b = c, the possible number of oscillating modes N L for longitudinal wavesis given by:

N L = 4πa3f 3

3v3

Because transversal waves have two possible polarizations holds for them: N T = 2N L.

5.6 Non-linear wave equations

The Van der Pol equation is given by:

d2x

dt2 − εω0(1 − βx2) dx

dt + ω20x = 0

βx2 can be ignored for very small values of the amplitude. Substitution of x ∼ eiωt gives: ω =12 ω0(iε ± 2

1 − 12 ε2). The lowest-order instabilities grow with 12 εω0. While x is growing, the 2nd

term becomes larger and decreases the growth. Oscillations on a time scale ∼ ω−10 can exist. If x isdeveloped as x = x(0) + εx(1) + ε2x(2) + · · · and this is substituted we have, besides periodic, secular terms ∼ εt. If we assume there exist some timescales τ n, 0 ≤ τ ≤ N with ∂ τ n/∂t = ε

n

and if we putthe secular terms 0 we get:

d

dt

1

2

dx

dt

2+ 12 ω

20x

2

= εω0(1 − βx2)

dx

dt

2

This is an energy equation. Energy is conserved if the left-hand side is 0. If x2 > 1/β , the right-handside changes sign and an increase in energy changes into a decrease of energy. This mechanism limitsthe growth of oscillations.

The Korteweg-De Vries equation is given by:

∂u

∂t + au

∂u

∂x non−lin

+ b∂ 3u

∂x3 dispersive

= 0

This equation is for example a model for ion-acoustic waves in a plasma. For this equation, solitonsolutions of the following shape exist:

u(x − ct) = 3ccosh2( 12

√ 2(x − ct))


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Chapter 6

Optics

6.1 The bending of light

For the refraction at a surface holds: ni sin(θi) = nt sin(θt) where n is the refractive index of thematerial. Snell’s law is:

n2n1

= λ1λ2

= v1v2

If ∆n ≤ 1, the change in phase of the light is ∆ϕ = 0, if ∆n > 1 holds: ∆ϕ = π . The refraction of light in a material is caused by scattering at atoms. This is described by:

n2 = 1 + nee

2

ε0m

j

f jω20,j − ω2 − iδω

where ne is the electron density and f j the oscillator strength , for which holds:

jf j = 1. From

this follows that vg = c/(1 + (nee2/2ε0mω2)). From this the equation of Cauchy can be derived:

n = a0 + a1/λ2. More general, it is possible to develop n as: n =n

k=0

akλ2k

.

For an electromagnetic wave in general holds: n =√

εrµr.

The path, followed by a lightray in material can be found with Fermat’s principle :

δ

2 1

dt = δ

2 1

n(s)

c ds = 0 ⇒ δ

2 1

n(s)ds = 0

6.2 Paraxial geometrical optics

6.2.1 Lenses

The Gaussian lens formula can be deduced from Fermat’s principle with the approximations cos ϕ = 1and sin ϕ = ϕ. For the refraction at a spherical surface with radius R holds:

n1v −

n2b =

n1

−n2

R

where |v| is the distance of the object and |b| the distance of the image. Applying this twice resultsin:

1

f = (nl − 1)

1

R2− 1

R1

where nl is the refractive index of the lens, f is the focal length and R1 and R2 are the curvatureradii of both surfaces. For a double concave lens holds R1 < 0, R2 > 0, for a double convex lens holdsR1 > 0 and R2 < 0. Further holds:

1

f =

1

v − 1

b

D := 1/f is called the dioptric power of a lens. For a lens with thickness d and diameter D holds ingood approximation: 1/f = 8(n

−1)d/D2. For two lenses placed on a lin

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